Estimator Campuran Spline Truncated dan Deret Fourier Pada Regresi Nonparametrik Multirespon

Nurcahayani, Helida (2022) Estimator Campuran Spline Truncated dan Deret Fourier Pada Regresi Nonparametrik Multirespon. Doctoral thesis, Institut Teknologi Sepuluh Nopember.

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Abstract

Analisis regresi nonparametrik multirespon menjadi solusi pada kasus data riil yang melibatkan lebih dari satu variabel respon saling berkorelasi dengan kurva regresi yang tidak diketahui bentuknya atau tidak mengikuti suatu pola tertentu. Selanjutnya, pada regresi nonparametrik multirespon yang melibatkan banyak variabel prediktor dimungkinkan terdapat perbedaan pola hubungan antar variabel prediktor dengan variabel respon sehingga penelitian ini mengembangkan model estimator campuran pada regresi nonparametrik multirespon. Penelitian ini mengembangkan model estimator campuran spline truncated dan deret Fourier yang mampu mengatasi pola hubungan variabel prediktor terhadap respon yang berubah-ubah pada sub-sub interval tertentu dan sebagian yang lain mengikuti pola yang berulang mengikuti garis trend pada interval tertentu. Tujuan teoritis dari penelitian ini adalah mengestimasi kurva regresi dari model, mendapatkan estimasi matriks varian kovarian serta memilih parameter penghalus yang optimal. Kajian simulasi dilakukan untuk mengevaluasi kinerja model yang diusulkan dan kajian terapan dilakukan sebagai bentuk pengaplikasian model pada data riil. Estimasi kurva regresi didapatkan melalui estimasi dua tahap yaitu (1) Penalized Weighted Least Square dan (2) Weighted Least Square dengan hasil estimasi mempunyai sifat linier terhadap observasi. Untuk mengakomodir korelasi antar respon, dilakukan estimasi matriks varian kovarian error model menggunakan metode Maximum Likelihood Estimation. Metode yang digunakan untuk memilih parameter penghalus, titik-titik knot, dan parameter osilasi yang optimal adalah Generalized Cross Validation. Kajian simulasi dengan variasi ukuran korelasi dan jumlah pengamatan memberikan kesimpulan bahwa semakin besar ukuran korelasi dan jumlah sampel maka semakin baik pula model yang diperoleh. Aplikasi data riil diterapkan pada empat variabel prediktor dan empat variabel respon yaitu data indikator Indeks Pembangunan Manusia (IPM) di Provinsi Jawa Timur pada 2019. Berdasar kriteria minimum GCV, hasil pemodelan data riil menunjukkan bahwa model campuran dengan dua prediktor didekati dengan fungsi spline truncated dan dua prediktor lainnya dengan fungsi deret Fourier menghasilkan hasil yang lebih baik daripada model tanpa estimator campuran dengan indikator kebaikan R2 sebesar 99,891% dan MSE = 0,755.
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Multiresponse nonparametric regression analysis has become a solution in the case of real datasets involving more than one response variable that is correlated with an unknown regression curve or does not follow some prespecified functional form. Furthermore, in multiresponse nonparametric regression that involves several predictor variables, it is possible that each of them has a different relationship pattern with the response variable. Therefore, the multiresponse nonparametric regression analysis with combined estimators is required. This research develops a combined truncated spline and a Fourier series model that have the ability to handle shifting behavior on the data at certain subintervals while some others follow a repeated pattern at specific intervals. The theoretical objectives of this study are to estimate the regression curve, obtain an estimation of the covariance variance matrix, and choose the optimal smoothing parameter. A simulation study was conducted to evaluate the proposed model performance and an applied study was carried out as the application of models to the real dataset. The regression curve estimation was obtained through two-stage estimation: (1) Penalized Weighted Least Square and (2) Weighted Least Square, which results in linearity properties with the observations. To accommodate the correlation between responses, an estimation of the variance-covariance error matrix model was performed using the Maximum Likelihood Estimation method. Later, the selection of optimal smoothing parameters, knot points, and oscillations parameters was obtained using the Generalized Cross Validation method. From the simulation studies with different sizes of the correlation and sample size variation, it is concluded that the best model was obtained through a larger sample size and correlation value. The implementation in real datasets was applied to four predictor variables and four response variables of Human Development Index (HDI) indicator data across East Java Province, in 2019. Based on the minimum GCV criteria, the results of real data modeling show that a combined model with two predictors approximated by a truncated spline function and the other two predictors by Fourier series function yielded better results than the model without the combined estimator with an indicator of R2 equal to 99.891% and MSE = 0.755.

Item Type:	Thesis (Doctoral)
Uncontrolled Keywords:	deret Fourier, estimator campuran, multirespon, regresi nonparametrik, spline truncated, Fourier series, combined estimator, multiresponse, nonparametric regression
Subjects:	H Social Sciences > HA Statistics > HA31.3 Regression. Correlation Q Science > QA Mathematics > QA278.2 Regression Analysis. Logistic regression Q Science > QA Mathematics > QA404 Fourier series
Divisions:	Faculty of Science and Data Analytics (SCIENTICS) > Statistics > 49001-(S3) PhD Thesis
Depositing User:	Helida Nurcahayani
Date Deposited:	18 Feb 2022 02:20
Last Modified:	18 Feb 2022 02:20
URI:	http://repository.its.ac.id/id/eprint/94231

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